Theorem 3dimlem1

Description: Lemma for 3dim1 . (Contributed by NM, 25-Jul-2012)

	Ref	Expression
Hypotheses	3dim0.j	⊢ ∨ = ( join ‘ 𝐾 )
	3dim0.l	⊢ ≤ = ( le ‘ 𝐾 )
	3dim0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	3dimlem1	⊢ ( ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3dim0.j	⊢ ∨ = ( join ‘ 𝐾 )
2		3dim0.l	⊢ ≤ = ( le ‘ 𝐾 )
3		3dim0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		neeq1	⊢ ( 𝑃 = 𝑄 → ( 𝑃 ≠ 𝑅 ↔ 𝑄 ≠ 𝑅 ) )
5		oveq1	⊢ ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
6	5	breq2d	⊢ ( 𝑃 = 𝑄 → ( 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ↔ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )
7	6	notbid	⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ↔ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) )
8	5	oveq1d	⊢ ( 𝑃 = 𝑄 → ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) )
9	8	breq2d	⊢ ( 𝑃 = 𝑄 → ( 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ↔ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) )
10	9	notbid	⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ↔ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) )
11	4 7 10	3anbi123d	⊢ ( 𝑃 = 𝑄 → ( ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ) )
12	11	biimparc	⊢ ( ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ) ∧ 𝑃 = 𝑄 ) → ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ∧ ¬ 𝑇 ≤ ( ( 𝑃 ∨ 𝑅 ) ∨ 𝑆 ) ) )